A261711: Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0.

OEIS matches ?

• A155543: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A347478: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A000302: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A261711: triangle, 49 terms agree, 13 full rows row lengths from the atlas (tabf) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 992 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:      1;
  1:      4;
  2:     15,     1;
  3:     56,     8;
  4:    209,    46,     1;
  5:    780,   232,    12;
  6:   2911,  1091,    93,    1;
  7:  10864,  4912,   592,   16;
  8:  40545, 21468,  3366,  156,  1;
  9: 151316, 91824, 17784, 1200, 20;

The row polynomials satisfy P(n,x) = 4*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4.

P[0, x_] := 1;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[4*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = 4*v[i - 1] - (1 - x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Triangle begins:
  0:      1;
  1:      4;
  2:     15,     1;
  3:     56,     8;
  4:    209,    46,     1;
  5:    780,   232,    12;
  6:   2911,  1091,    93,    1;
  7:  10864,  4912,   592,   16;
  8:  40545, 21468,  3366,  156,  1;
  9: 151316, 91824, 17784, 1200, 20;

The row polynomials satisfy P(n,x) = 4*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4.

P[0, x_] := 1;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[4*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = 4*v[i - 1] - (1 - x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-791d6802d1e7f8a6

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image affine, shift 0 e-791d6802d1e7f8a6

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers paper_theorem e-0801036cdfff24d2 root: theorem check

      Parent-row provers of polytool polytool ?

    • parent_provers paper_theorem e-0d3631d0f359c9ce root: theorem check

      Parent-row provers of polytool polytool ?

    • gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): delannoy
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang_derivative_lag e-15c233e4385971d1 root: theorem check

          Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

        • preserver_image reversal, shift 0 e-266d85ca03a298fa

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): delannoy
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang_derivative_lag e-15c233e4385971d1 (expanded above)
            • basis_map A008288 e-e771e35dc9ee3fbd

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
            • basis_map A008288 e-33dba7e79fe9caa8

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
        • ↵ basis_map A008288 e-e771e35dc9ee3fbd (expanded above)
        • ↵ basis_map A008288 e-33dba7e79fe9caa8 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003